An elementary course of infinitesimal calculus . nate of the centre. If C, C be the centres of two circles of the system, the linejoining their intersections bisects GC at right angles. Hence thepoints of ultimate intersection of any circle with the consecutivecircle are the extremities of the diameter which is perpendicularto the line of centres. The envelope therefore consists of twostraight lines parallel to the line of centres, at a distance equalto the given radius. Fig. 120. Ex. 2. A straight line including, with the coordinate axes,a triangle of constant area (k^). If AB, AB be two posi
An elementary course of infinitesimal calculus . nate of the centre. If C, C be the centres of two circles of the system, the linejoining their intersections bisects GC at right angles. Hence thepoints of ultimate intersection of any circle with the consecutivecircle are the extremities of the diameter which is perpendicularto the line of centres. The envelope therefore consists of twostraight lines parallel to the line of centres, at a distance equalto the given radius. Fig. 120. Ex. 2. A straight line including, with the coordinate axes,a triangle of constant area (k^). If AB, AB be two positions of the line, intersecting in F,the triangles APA, BPB will be equal, whence = Hence, ultimately, when AA is infinitely small, P will be themiddle point of AB. If x, y be the coordinates of P, and w theinclination of the axes, we have, then, OA = 2x, OB = 2y, andtherefore 2xy sin m^Ii?. The envelope is therefore a hyperbola having the coordinate axesas asymptotes. Mg. 121 illustrates the case of m= Jtt. 165-156] CURVATURE. 415. 156. General Method of finding Envelopes. The equation of any curve of the system heing cl,(x,y,a) = 0 (1), where a is the parameter, then at the intersection with another curve (2), (3). ^ (a?, y. a) = 0 we have, evidently, ^ fe y, a) - ^ {«, V) ^Qa —a When the variation a — a of the parameter is infinitelysmall, this last equation takes the form ^4>{x,y,a) = 0 (4), where 9/9a is the symbol of partial differentiation withrespect to a. See Art. 45. The coordinates of the point, or points, of ultimateintersection are determined by (1) and (4) as simultaneousequations, and the locus of the ultimate intersections is to befound by elimination of a between these equations. 416 INFINITESIMAL CALCULUS. [CH. X Ex. 1. The circles considered in Art. 155, Ex. 1 may berepresented by {x-af + y = a? (5). Differentiating with respect to a, we find x-a=0 (6). Eliminating a between (5) and (6) we get y = ±a (7), the envelope required. Ex. 2. If
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